Recreations in Mathematics
Series Editor
David
Singmaster
1. Hugh ApSimon
Mathematical byways in Ayling, Beeling, and Ceiling
2. John D. Beasley
The ins and outs of Peg Solitaire
3. Erno Rubik, T a m b Varga, Gerzson K b i , Gyorgy Marx, and T a m b Vekerdy
RubikS cubic compendium
4. Edward Hordern
Sliding piece puzzles
I
Oxford
New York TokyoOXFORD UNIVERSITY PRESS
1986
Oxford University Press, Walton Street. Oxford OX2 6DP Oxford New York Toronto
Delhi Bombny Calcuna Madrm Karachi Petaling laya Singapore Hong Kong Tokyo Nairobi Dares Salaam Cape Town Melbourne Auckland
and arsociated companies in Beirut Berlin Ibadan Nicosia
Oxford is a trade mark of Oxford University Press Published in the United Stales
by Oxford University Press, New York
0
Edward Hordern, 1986AN rights reserved. No port of this publication may be reproduced, stored in a retrieval system, or transmitted. in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press
British Library Cataloguing in Publication Data Hordern, Edward
Sliding piece puzzles.
-
(Recreations in mathemotics; No. 4)I . Mathemat~cal recreations I. Title II. Series 793.7'4 QA95 ISBN 0-19-853204-0
Library of Congress Cataloging in Publication Data Hordern, Edward.
Sliding piece puzzles.
(Recrearions in mathematics; 4) Bibliography: p.
Includes index.
1. Mathematicol recreations. 2. Puzzles. I. Title. 11. Series.
QA95.H67 1986 793.7'4 86-18204 ISBN 0-19-853204-0
Set by Hope Services, Abingdon, Oxon Printed in Great Britain by
Butler & Tanner Ltd,
Frome, Somerset
To Christopher, Mandy, Fern, and
Sam
It's all very well for an hour or two But people might think it a bore
When after six weeks the solution one seeks Is just as far off as before.
The thing I've abused, Oh! the language I've used Half out of my mind I have been
And I'm rueing the day that they put in my way That horrible puzzle 'Fifteen'.
Foreword
David Singmaster
The sliding piece puzzle is one of the most popular of all mathematical recreations. A close relativ-the famous Rubik's Cube-in 1980 spawned one of the two greatest puzzle crazes of all time. What is not so well known is that the other great puzzle craze- exactly a hundred years earlier involved a sliding piece puzzle called the 'Fifteen Puzzle'. In the ensuing century, there has been a remarkable numher of different types that have appeared.
I first became aware of the profusion of such puzzles when I met Edward Hordern about five years ago and he gave me a copy of his booklet 150 Sliding Block Puzzles. When this Series was first proposed, I immediately suggested that Hordern should update his booklet and this hook is the result. He has significantly increased the numher of sliding block puzzles described and has somewhat broadened the scope of the work to include shunting and switching puzzles and many other types of sliding piece puzzle, so that this hook describes over 250 puzzles.
Edward Hordern, like many puzzle enthusiasts, has come to puzzles by a roundabout route. After some years in advertising and accounting, he started a small manufacturing firm which has prospered and gives him some time and the resources to pursue his hobbies. About fifteen years ago, he began to collect mechanical p u z z l e ~ t h a t is, puzzles where a physical object must he manipulated in some way (though jigsaws are generally excluded). His collection is now one of the finest ever assembled, comprising some 8000 items, including many from the 19th century. Among these are about 750 sliding piece puzzles. He has an especial affinity for such puzzles and cheerfully spends hours, sometimes even years, finding better solutions for them. With his wealth of experience, I doubt if anyone can surpass his speed in solving a new puzzle. Most of the solutions given here have been found by him and many of these are notable improvements on previous solutions. In this hook, Hordern shares his fascination and passion with the rest of us.
Preface
This is not a book for readers!
It is a collection of more than 270 sliding piece puzzles with the pieces provided for solving them. In the back of the book there is an envelope containing 'push out' pieces which can he used to solve some 230 of the puzzles. If the pieces should get lost or mislaid there is a 'back up' page of designs in Chapter 1 giving all shapes and quantities required of each piece.
There is a peculiar fascination in pushing blocks around a board- trying to get one to a particular position or attempting to form a certain pattern. Once started, people find them hard to put down. Often the only reason they don't pick them up in the first place is the fear of failure and the (quite unreasonable) thought that they might be made to look a fool or intellectually inferior. But the ability to solve sliding piece puzzles, like crosswords, has little to do with intelligence: some people find them easier to do than others. Just as there is a 'knack' to solving crosswords, so there is with sliding piece puzzles. Being able to think clearly and look ahead will help greatly. While the easier puzzles can often be solved by trial and error, the harder ones can rarely be solved this way. Considerable intuition and perseverance will be needed to crack the tough ones.
Whatever one's aptitude, the main purpose of this book is to provide entertainment. There is something for everyone: some puzzles are unbelievably easy, others so hard that few will solve them. And there are all those in between that are designed to give hours-perhaps even months--of fun.
Oxfordshire
E.H.Acknowledgements
Many people from all over the world have helped in the preparation of this book. They fall broadly into three categories:
(a) Those who have given me help, information and advice.
(b) Those who have let me have details of puzzles in their collections. (c) Those who have helped with some of the solutions.
Many people have let me have information and help on many puzzles and other information in this book and I thank tbem all. These include David Singmaster (England), Martin Gardner (USA), Nob Yoshigahara (Japan), James Dalgety (England), Dick Hess (USA), Trevor Trurau (England), Stanislav Tvrdik (Czechoslovakia), Stewart Coffin (USA), David Pritchard (England), George Jelliss (England), and Dan Feldman (Israel). Many thanks to these and others who I have not mentioned.
I am most grateful to Jerry Slocum (USA), the late Eileen Scott (England), Abel Garcia (USA), and Will Strijbos (Netherlands) for having allowed me to include details of many antique (and modern) sliding block puzzles in their collections.
Last but by no means least I would thank those who have either improved on my solutions or who gave the best solutions in the first place: members of the Academy of Recreational Mathematics of Japan, Nob Yoshigahara (Japan), Kiyoshi Takizawa (Japan), Charles Levitt (South Africa), H. Moriya (Japan), T. Saito (Tokyo University), Len Gordon (USA), Ian Pedder (England), and K. Okada (Japan).
The following puzzles have been reproduced with the permission of their publishers:
Academic Press 1nc.-'Century and a Half' from Winning ways, Vol. 2 by G . Berlekamp, J. Conway, and R. Guy.
George Allen & Unwin-'The Short Siding' from Maths is Fun by J. Degrazia.
A. S. Barnes & Co.-'Bull's Eye', '5, 11, 15, 19 Block Puzzles' from 100
puzzles, how to make and solve them by A. Filipiak.
Dover Publications 1nc.-Introduction by P. van Note from The 8th book of
Tan by S. Loyd.
Electrical-Electronic Press-'George Orwell Puzzle' by T. Truran from
xii Acknowledgements
W. H. Freeman & Co.-'The Eight Block Puzzle' from Sixth book of mathematical games from Scient$c American by M . Gardner.
Harper & Row-'Bull Pen' from Games &puzzles you can make yourself by H. Weiss (Thomas Y. Crowell). Copyright
0
1976 by H. Weiss.Harper & Row and A. & C. Black Publishers, Ltd.-'Lots of Luck' from
What's the big idea? by D. Rubin-illustrations by R. Jones (J. B. Lippincott). Copyright
0
1979 by D. Ruhin.Michael Joseph-'Two Sliding Puzzles' from Games &puzzles for addicts by
R. Millington.
Simon & Schuster-'Rate your mind pal' from The Scientific American book of mathematical puzzles and diversions by M. Gardner.
Times Mirror Magazines 1nc.-'Strategy' (1940) and 'Change the Seasons' (1942) from Popular Science.
United Media and World Almanac Publications-'Jeepers' from Realpuzzle book by D. Rubin.
A. P. Wart Ltd.-'Set Up' from Figure it out by D. St. P. Barnard.
Contents
1. About the puzzles
T h e '15' Puzzle. Scope. Difficulty ratings. Taxonomy. Identification. Moves. Solutions. Hints and tips. Parity. Making t h e puzzles
2. History of the sliding block puzzle
'15' and '14-15' puzzles. O t h e r puzzles
3. Random arrangement sliding block
puzzles, A1-A10
4. Sliding block puzzles with uniform
pieces, El-B60
5.
Sliding block puzzles with rectangular
pieces, C1-C76
6. Sliding block puzzles with non-convex
pieces, Dl-D50
7. Restricted route sliding block
puzzles, El-E35
8. Sliding block puzzles with special shaped
pieces or where rotation is allowed, F1-F13
9.
Three-dimensional sliding block
puzzles, GI-G5
10. Soko (warehouse) puzzles, HI-H10
11.
Railway shunting puzzles, 51-513
Solutions
Appendix A: A mathematical note
by D a v i d S i n g m a s t e rAppendix B: A List of patents
Appendix
C: References and bibliography
Index
ABOUT THE PUZZLES
The
'15' puzzle
It all started just over 100 years ago; the world's first puzzle craze. The second one-almost exactly a hundred years later-was the Rubik's Cube phenomenon. There cannot be many educated people in the world today who have not heard of a Rubik's cube, nor handled one.
And so it was in 1879 and 1880, that a small square box carrying 15 little square blocks of wood captured the imagination of the world and caused at least as much of a stir as the Rubik's Cube. It was called the '15' Puzzle.
The 15 numbered blocks were t o be put into the (4 x 4) square box at random. All the solver had to do was to slide them one at a time, without lifting or turning, so as to put them in order: 1-4 in the top row, and so on down to 13-15 in the last row, followed by the vacant space (See Figure 1).
v
/Figure I
The great fascination of the puzzle was that sometimes it was solved very easily and at other times it would not yield to any amount of sliding of the blocks. It was possibly this element of uncertainty in something so apparently simple that made the puzzle so irresistible. It does not appear reasonable to the simple human mind that sometimes it
2 Sliding Piece Puzzles
should work out and not others. One's natural instinct is to say chat either it should always succeed or not at all.
From this explosive beginning has developed the multitude of sliding piece puzzles of all shapes and sizes, from the absurdly easy to the incredibly difficult.
Scope
There are four kinds of sliding piece puzzle, the first three of which are covered in this book:
(i) Sliding block puzzles-in which each piece can move indepen- dently of the others;
(ii) Soko (warehouse) puzzles-in which one piece must push all the others;
(iii) Railway shunting puzzles-in which one or two pieces must push or pull all the others;
(iv) Sliding puzzles with plungers o r levers-in which pieces move in groups.
Category (iv) has been omitted because readers are unlikely to be able to construct them, and although they achieved a popularity a few years ago, they are largely unobtainable now. A selection is shown in Plates X and XI.
What all sliding piece puzzles have in common is (a) pieces that slide, without Lifting or jumping, from one position t o another, and (b) a space into which to slide the pieces.
The famous 15 Puzzle belongs to the sliding block category. There have been considerably more of this type than of all the others put together. They are certainly the more interesting to solve and their relative ease of manufacture has ensured their continued popularity. Over the last 100 years so many sliding block puzzles have been produced and then forgotten about, and so few seem to be available today, that the idea came of assembling everything of interest and putting it into one book. However, as research progressed, it became evident that there was far too much material, and some other criterion was needed. It was decided not to include any puzzle unless it has at some time been manufactured or published in some forn-published material includes books, magazines, scientific papers and patents. Certain puzzles have been included because of their historical value rather than their interest as puzzles; others, interesting as they may be, have had to be excluded, usually because of their similarity to puzzles already included. All the puzzles listed have
About thepuzzles 3 been made in one form or another (most are in the author's collection), except for those where only a book, magazine or patent is shown as the source. It is not known whether these puzzles were ever produced. The sources of the books and magazines is given in Appendix C and a list of patents in Appendix B. The country of origin of all other puzzles is given alongside their titles. Where no country is shown, the puzzle emanated from the UK.
The chapters have been arranged accordmg to the type of puzzle. Some people may find one type of puzzle more interesting than another. Consequently, when attempting the sliding block puzzles in Chapters 3 to 7, the reader is recommended to start with the puzzles in Chapter 5, which may have a more general appeal, before attempting those in Chapters 3 and 4.
It is often impossible to tell just by looking at it whether a puzzle is more worthwhile than others, and therefore a list of 'preferred' puzzles is given at the beginning of each chapter.
Difficulty
ratings
As is so often the case with puzzles, some that look very easy to solve are in reality very difficult and vice versa. A grading system, rating the degree of difficulty of every puzzle, has been introduced to help the reader:
*
easy; warm-up exercise;* *
standard;* % a tricky;
* * * *
difficult;*****
exceptionally difficult;******the ultimate challenge.
Beginners are urged to start with puzzles rated
*
and* *
and work their way upwards. Otherwise disillusion can set in quite quickly! Even readers with some experience of this type of puzzle are urged not t o start with the* * * *
,d* * * * *
puzzles, but t o progress to them gradually. Be warned! Experts can take days or weeks to solve the most difficult ones. The author took four years to discover the insight leading to one solution-a puzzle he had previously thought impossible.The object in some of the puzzles (mostly in Chapters 3 and &a few in Chapter 7) is to solve them in a specified number of moves. Just to solve the puzzle in unlimited moves is too easy. Therefore a second rating in brackets: '. . . in minimum moves' gives the rating
4 Sliding Piece Puzzles
for solving the puzzles in the required number of moves. It is this rating that the reader should follow.
The ratings are necessarily subjective and one person's view, but it is hoped they will provide a reasonable guide.
In the taxonomy of mechanical puzzles there are more than a dozen principal classes of puzzle. Examples include: 'disentanglement' (wirelwire and string puzzles), 'dexterity' (rolling ball puzzles) and 'route finding' (mazes). One
of
these categories is called 'sequential movement puzzles'.The sequential movement category cormprises three sub-classes: (i) Rotational puzzles (includes the famous 'Rubik's Cube'); (ii) Sliding piece puzzles;
(iii) Peg jumping puzzles (includes the well-known European 'Solitaire' puzzle).
This book is about the middle category. Sliding piece puzzles can be further divided into:
(a) Sliding block puzzles (Chapters 3-9); (b) Warehouse (Soko) puzzles (Chapter 10); (c) Railway shunting puzzles (Chapter 11).
(d) Sliding puzzles with plungers or levers (Plates X and XI);
Identification
What is the definition of a sliding piece puzzle? A sliding piece puzzle consists of a number of pieces of any shape(s) enclosed within a defined area (or confined space), in which the purpose is either to rearrange the pieces into a predetermined order or to get a particular piece into a specified position. This is accomplished by sliding the pieces into areas not occupied by other pieces. In sliding block puzzles, all the movable pieces are capable of independent movement, but in most cases are not allowed to rotate; in warehouse (Soko) puzzles there is the restriction that none of the pieces can move except one, which must push (never pull) all the other pieces one at a time into their correct positions; in railway shunting puzzles there is the restriction that none of the pieces can move except one or two, which must push or pull the other pieces, singly or in groups, along specified routes into their correct order or places; and in the
About the puzzles 5 plunger/lever type puzzles the pieces can only be moved in groups by means of plungers or levers.
The difference between jumping peg puzzles and sliding piece puzzles is obvious, as the latter allows no jumping. However, the difference between the plungerllever puzzles and rotational puzzles can appear to be rather obscure. The essential difference is in the existence of a space. A Rubik's Cube has no space and there should be no difficulty in seeing that it does not belong in the sliding piece category. However, the plungerilever puzzles could be mistaken for rotational puzzles since they apparently have no space. In point of fact, the plunger type puzzles do have spaces even though they may be hidden from view or occupied (temporarily) by plungers, levers andlor springs. Although these spaces are outside what might be termed the 'working area' of the puzzle, they exist just the same. Pieces are moved into these spaces while other operations are performed. Therefore, in the strict sense of the word, they come into the definition of sliding piece puzzles.
Moves
Definitions of moves for Soko (warehouse) puzzles and railway shunting puzzles are given at the beginning of Chapters 10 and 11 respectively.
What constitutes a move in a sliding block puzzle? There are several possible definitions and, unfortunately, authorites cannot agree on a standard definition.
There are four possible alternatives:
1. Slide one piece only in any direction or combination of directions. The piece may be slid any permissible distance without lifting and (unless specifically allowed) without rotating. (Under this definition a piece can move around a corner; see Figure 2.)
2. Slide one piece only in any one direction. The piece may be slid any permissible distance without lifting or rotating (see Figure 3).
3. Slide any number of pieces together as a group in any one direction. Pieces may be slid any permissible distance without lifting or rotating (see Figure 4).
4. Slide one piece in any (orthogonal) direction, one unit ( = the same distance as the dimension of the smallest piece; see Figure 5). Definition (i) is the preferred definition for several reasons. The majority of puzzles produced have specified or intimated this definition where solutions have been given. It is the accepted
6 Sliding Piece Puzzles
definitions of most (but not all) puzzle experts and collectors. The normal method of moving pieces will be with one's forefinger. It is logical that a move should start with the placing of the finger on the piece and end when it is taken off. It is irrelevant whether it has negotiated a corner in the meantime. If definition (ii) is used, an ambiguity occurs in situations where it is possible to move diagonally.
Figure 2 Figure 3 Figure 4
Figure 5 Figure 6
In Figure 6 the small square has the possibility of moving in two different ways to the spot marked X. If one allows only orthogonal movement, it would count as two separate moves under definition (ii); a diagonal move, on the other hand, would obviously only count as one. Under definition (i) no such confusion arises; it counts as one move however the piece is moved.
It might seem incredible that anyone could seriously propose definition (iii), which is the most awkward and unwieldy of them all. But in the 1930s a magazine specializing in games used it as their definition in an article about sliding block puzzles. More recently a manufacturer of a sliding block puzzle instructed solvers to use it too. At the end of the description of each puzzle, the number of moves needed to solve it is shown. From Chapter 5 onwards it is given in the form: 'Solution: 76 moves (87)'. The first figure refers to the number
About thepuzzles 7
of move required under definition (i) and the second, which is always in brackets, refers to definition (ii). In the solutions section, where different from each other, solutions for both definitions are given. The only exception to this is for some of the difficult puzzles at the end of Chapter 6, where only solutions for definition (i) are given. The reason that definition (ii) solutions are omitted is that confusion may arise as to whether a diagonal move should count as one move or two.
Solutions
The solutions given at the end of the book are the shortest currently known. A handful of these have been proved to be minimal by computer. The majority of the remainder are thought to have the minimum moves. Some of the really hard puzzles are probably not minimal, but just to solve them is satisfaction enough.
The majority of the solutions have been done by hand by the author, and many have been improved (shortened). For example the original solution t o Ma's Puzzle (Dl) was given as 61 moves, but it is easily solvable in 23 moves. Similarly, the original solution to Get my Goat (C2) had 46 moves--this has now been reduced to 28. These unnecessarily long solutions are not confined to the older puzzles. Modern puzzles sold in the last few years with incredibly long solutions include Lost Pygmies (C70), whose given solution has 274 moves but can be done in 130, and Little Hippo (C63), where the given solution has 231 moves but needs only 84 moves. These are just a few of many. The original solutions to the above puzzles are so much longer than necessary that it is hard to believe that the solvers were really trying!
The complete answer t o getting the shortest solutions is to use a computer. After much deliberation it was decided not to include a section on programming computers to solve sliding block puzzles, because it would take all the fun out of solving the puzzles. It would be like playing a game of chess and getting a computer to play for you--or doing a crossword and looking up the answers before one has looked at the clues.
Hints and tips
Several puzzles have been included that are not so much puzzles as 'warm-up' exercises for the more difficult ones. Jl is essential to
8
Sliding
Piece Puzzlesmaster the principles involved, because they recur frequently as part of other puzzles. For all the puzzles in Chapters 3 and 4, it is important to know how to exchange the positions of two adjacent pieces within a confined space. Puzzle B1 is the required 'exercise' (puzzles B19 and B20 are a good extention to this principle). Although more difficult, B46 is another good exercise in the manipulation of uniform pieces. When rectangular pieces are introduced, the first exercise is to solve C l . This will illustrate the principle that what at first sight appears to he impossible can he solved with a little perseverance.
Puzzles that have pieces with assorted sizes and shapes present quite different problems, and the most helpful pieces are always the small squares. These usually have to precede or follow the large square in order to facilitate its progress--especially when it comes to turning a corner. Good puzzles to illustrate the point are C19 and C52.
L-shaped pieces add a whole new (difficult) dimension and the best exercise is D4, which illustrates several principles.
For the more difficult puzzles, especially those in Chapter 6, it is quite useless sliding the pieces at random in the hope of getting somewhere. It is most important to develop a strategy. This should consist of a series of intermediary objectives along the following (much simplified) lines: piece x cannot be moved to m until piece y is moved out of the way; piece y is blocked by piece z , which must he moved first. . . Continuing on from this reasoning: piece x cannot get past piece y until pieces r, s, tare out of the way and pieces a and b are in their place. An example of tactical strategy would be to plan to get all the pieces into a certain order from which the puzzle can he solved. In many cases, it is the largest or most akwardly shaped piece (the key piece) that has to he moved to a specific location. This may involve rearranging almost all the other pieces of the puzzle before the key piece has moved at all. A further hint is to move L-shaped pieces together to form a rectangle and to note how they move easily round corners in some directions but not others.
If the more difficult puzzles are tackled on the basis that they are 'puzzles within puzzles' then they often readily yield to solution.
Parity
When the '15' Puzzle (see B10 in Chapter 4) was first produced, would-be solvers were intrigued to find that sometimes the puzzle
About the puzzles 9 came out very easily and at other times it appeared to he unsolvable. The first mathematical papers showed that exactly half of all the possible random start positions led to a solution (and half did not). Attention was then focused on the mathematical laws that governed problems of this sort: was there a general rule that governed all puzzles of this kind, and was there an easy method of identifying whether a particular start position was solvable? The answer to both questions was, of course, yes.
The law governing problems of this type is commonly referred to as 'parity', and performing a 'parity check' is the means of identifying solvable and unsolvable positions. 'Changing the parity' of a puzzle means changing it from an unsolvable to a solvable state (or vice versa). The parity check method given below applies to a rectangular hoard of any shape or size providing it has uniform pieces and one space the same size as a piece.
1. Exchange any two pieces so that one or both pieces go to their correct position. Exchanging pieces means physically lifting them from their place-not normally allowed in solving the puzzles.
2. Repeat the exchanges until all the pieces end up in their correct positions.
3. Count the total number of exchanges; if the number is even, the puzzle is solvable from its original position; if the number is odd, it is not solvable. (The maximum number of exchanges is always one less than the number of pieces).
There is a mathematical theorem which shows that it does not matter how or in what order the exchanges are carried out-they may even he repeated-this test will always give the correct answer.
An easy way of counting the exchanges in, for example, the '15' puzzle, is to count a 'cycle' of exchanges, as follows:
(a) Pick up piece No. 1 and put it in its correct place, removing the number already there.
(h) Repeat (a) with the new number.
(c) Continue until a number goes into a vacant space, which ends the cycle. Count the number of moves in the cycle. The number of exchanges is always one less than the number in the cycle.
(d) If there remain numbers not positioned correctly, start a new cycle and continue until all numbers are correctly placed. (With a little practice this method can he done in one's head).
The parity principle is used to effect (to confuse the solver) in certain puzzles in Chapters 3 and 4 and in one or two more puzzles in other chapters. It would appear from a pdrity check that these puzzles
10
Sliding
Piece Puzzlescan only be solved if parity can be changed. But is this possible? If so, how?
To go any further would be to reveal an important part of some of the solutions to these puzzles. Therefore, so as not to spoil the enjoyment for those who wish to tackle them without referring to the solutions, this discussion on parity is continued at the beginning of 1
the solutions section.
I
Making the puzzles
Two alternatives for makmg the puzzles are offered, depending on whether the reader wants 'instant' puzzles, or, with only a little time and effort, a set of superlor pieces.
Alternative
A
A set of 53 pieces on a pre-cut card is given in an envelope at the back of the book. The same pieces are reproduced in Figure 7 in case some get lost. The pieces in the envelope may be readily separated from each other and can be numbered or lettered, if required, so that the solutions can be followed. Where coloured pieces are needed, they can be lettered: B (blue), It (red) etc. For the railway shunting puzzles in Chapter 11, it is best to use the 2 x 1 rectangular pieces (where I1 or fewer are required). The direction of the engineslcoaches can be shown by marking them with an arrow. It is strongly recommended that a pencil is used to mark the pieces, as marks can then he erased easily and the pieces re-marked for a different puzzle.
The only further item that will be necessary is a hoard, tray, enclosure or box, in which to slide the pieces. The simplest board is made by drawing a thick black line with a felt tip pen on paper, card or any other suitable smooth surface to mark the outline of a particular puzzle. Obstacles and immovable blocks can also he marked in this way. It helps greatly in solving the puzzles if the pieces and the board are of different colours. For this reason, the pieces in the back of this hook have a colour tint.
When drawing a thick black line to mark the border for a selected puzzle, bear in mind that it needs to be slightly larger than the area of the pieces. In puzzles having a square or rectangular border an extra margin of one quarter of the smallest dimension of the smallest piece is recommended. For example, if the smallest piece is a square having a 12 mm side, then the extra margin around the edges should be
About the puzzles 11
12 Sliding Piece Puzzles
about 3 mm. A felt-tip pen can also he used to make a hoard consisting of circles connected by a series of lines.
By drawing the outlines of the puzzles on a suitable surface and using the pieces in the envelope provided, the reader will be able to tackle some 230 of the puzzles or m o r c s e e Table 2.
The pieces at the back serve a most useful purpose in providing 'instant puzzles'. But they suffer from two disadvantages. For reasons of space, they have had to be made far smaller than the ideal, and consequently they are rather fiddly to manipulate. Another drawback is that they have square corners, which have an annoying tendency to catch on neighbouring pieces when being slid from one place to another. Both these disadvantages can he overcome by making one's own pieces.
Alternative
B
Since most people shudder at the thought of making anything, especially if it sounds the least bit complicated, every effort has been made to allow readers to make their own pieces with the minimum effort. The pieces listed in Table 1 are reproduced full-size as Figures 8 and 9 in the colour plate section: Figure 8 is the basic set and Figure 9 the supplementary set (see Table 2). Just take four photocopies of
Table 1: Piece shapes and sizes
Shape Unit size"
a T shape h Long L c Large L d Medium L e Small L f Large Square g Small Square h Rectangle i Rectangle j Rectangle k Rectangle I Circle m 1 % hexagon
less two non-adjacent (1 x 1) corners less one (3 x 1) corner
less one (1 x 1) corner less one (1 x '/2) corner less one (?4 X %) corner
" It is recommended that each unit should be approximately 20 mm.
About thepuzzles 13 each set, stick each complete photocopy onto thick card or hoard and then cut out the shapes with scissors or a sharp knife. Half an hour or so with glue and scissors is all that is required. If no photocopier is available, tracings can be used instead.
The simplest hoard can he made in the same way as in Alternative A, hut it is more satisfactory to make proper trays, so that the pieces slide about within a defined area or 'enclosure'. Full design details for a set of wooden trays are given below, hut they can just as well he made by gluing pieces of wood or card onto a suitably smooth surface.
The basic set of piece shapes in Figure 8, made up into four sets, provides all the same pieces that are in the back of the book, and a few extra that are not needed. For example, only one T-piece is needed, whereas the photocopying produces four. Table 2 gives all the pieces that are required, so that time need not be wasted making unnecessary pieces. The same 230 puzzles (or more) can he attempted with the basic set as with the ones provided at the back.
A supplementary set of piece shapes is reproduced in Figure 9, and these consist of unusual shapes that are used only for a few puzzles, as well as extra pieces of the basic set needed for some of the larger puzzles. Four photocopies of this set are also required. When used in conjunction with the basic set, they will allow the reader to tackle a further 17 puzzles-a total of 247.
It should not he beyond the scope of the enthusiast to make their own pieces for F6-F8 and FlO-F12. The remaining puzzles not covered in Table 2 cannot he attempted as their hoards are far too complex or they are three-dimensional.
Those readers who wish to solve the maximum amount of puzzles yet make the minimum number of pieces are recommended to make the following 38 pieces: one T shape, four large
L
pieces, four large squares (2 x 2), sixteen small squares (1 X I), two rectangular (3x
I) pieces, and eleven rectangular (2 x 1) pieces-most of the basic set. The 230 puzzles shown in Table 2 can still be tackled if the squares are used instead of the circles.The best pieces of all will he made out of wood, plastic or other suitable material. The designs in Figure 8 and 9 can be photocopied and used as templates for cutting out the pieces. Whatever pieces are made, ensure that they have rounded corners--it really
is
worth the extra trouble and makes the difference between an ordinary puzzle and one that is a joy to manipulate. The radius of the curve should be one eighth to one tenth of the smallest dimension of the smallestTable 2: Piece sets and the puzzles they will do Complete set Pieces Basic set Supplementary set Basic
+
needed for provided [Figure 8 (X 4)] [Figure 9 (X 4)] supplementary Piece 247 puzzles in envelope (X 4) a T shape 1 1 1+ 4a 4 b Long L 2 1- 4h 4 c Large L 6 4 1+ 4 1+ 4b 8 d Medium L 4 2-ad
8 e Small L 4 14 4 f 2x2 4 4 1- 4 4 g 1x1 35 16 4- 16 5-
2ff 36 h 4x1 4 1- 4 4 i 3x1 2 2 1- 4b 4 j 2x1 11 11 3 + 12C 12 k 1Mxl 7 2- 8'~ 8 1 Circle 15 15 4 + 16' m '/Z hexagon 16 4- 16 - - --
111 53 15-
60 17 + 68 Surplus pieces - - 7 10 - - - - TOTAL PIECES 111 53 53 58 No. of puzzles possible 247 230 230 N/A " Includes three surplus pieces. Includes fwo surplus pieces. ' Includes one surplus piece. Includes four surplus pieces-note that there are left-handed and right-handed versions of this piece. With either the pieces provided in the envelope at the back of the book or the Basic Set [Figure 8 (x 4)] all the puzzles in the book are possible except A3,B22, B26, B40, C5, D5-D8, D49, E10, Ell, E15-17, E33, E35, F1LF13, GI-G5, H4, H8, H10.55, J6, J12,513. If the circular pieces are used as additional squares, a further six puzzles can be attempted: B22, B26, B40, H4, H8, J5. Puzzles possible with the basic and supplementary sets together [Figures 8 and 9 (x 4)): all the puzzles in the hook except E10, Ell, E15-E17, E33, E35, F3-F13, GI-G5, J12, J13. 0 Dc
16 Sliding Piece Puzzles
piece. The ideal dimensions of the pieces themselves are such that the smallest piece (square) should be slightly larger than the tip of one's forefinger. As in Figures 8 and 9 this would mean a unit size of about 20 mm-making the small square 20 x 20 mm. This could he reduced to about 15 X 15 mm, hut any smaller and the pieces become difficult to slide easily.
It is very helpful in solving the puzzles if the pieces have different colours: all the small squares one colour, all the large squares a second colour, all the vertical rectangles a third colour, all the horizontal rectangles a fourth colour, etc. This makes it much easier to visualize what pieces are where and where they should go. The author has often used Lego pieces, which come in a variety of shapes (including L shapes), colours and sizes, for just such a reason. They are unsuitable in many respects (sharp corners, too small) hut are instantly available and their colours make solving difficult puzzles that much easier.
Drawing lines on a card to delineate the boundaries of the playing area is all very well but it is far from ideal. There is a world of difference between pushing pieces around a board that consists of a black line on a piece of paper, and a well-made sliding hlock puzzle with a proper raised border, where the pieces slide smoothly and do not catch on one another. If the reader is making his own pieces from wood or some other material, then it should not be too difficult to construct a tray or two with edges. It is suggested that two trays and six 'filler' blocks are constructed. With these it will be possible to adapt one or other of the trays to any of the 'board sizes' for the puzzles in Table 2. The tray sizes should theoretically he 10 x
7
units and 6 x 6 units (1 unit = the small square), but they will need to he slightly larger to accomodate the movement of the pieces. If the corners of the pieces are nicely rounded then the extra margin at the border of the trays should he about one eighth of the smallest dimension of the small square-about 2%-3 mm if the small square is 20 x 20 mm. If the corners are not rounded, then this margin should he doubled. The theoretical dimensions of the 'filler' blocks in units are: 10 x 1,6
X 1, 5 x 1, 4 X 1, 3 x 1, 2 x 1. The long dimension of each should he slightly more than the theoretical so as to take up the extra space created by the margin. This is especially important with the 10 X 1 and 6 X 1 blocks, which will need to he 'jammed' between the two sides of the puzzle (not necessarily along an edge). Pieces can also be used as filler blocks where necessary. By jamming two of the filler blocks, one crosswise between the sides andAbout the puzzles 17 one lengthwise between the top and the first filler block, it is possible to construct almost any size of hoard.
The base of the tray, on which the pieces are to slide, should he as smooth as possible. If made of wood, the surface should be sanded and either waxed, polished or lacquered.
It is a good idea, but not essential, to make lids for the trays so as to prevent pieces falling out or getting lost.
Thus with only two trays, six filler blocks and a choice of 53 or 111 pieces one has all the equipment needed to undertake some 230-247 puzzles.
The best-made puzzles that the author has seen are those made by Minoru Abe (see Plate VII bottom left-also puzzles D24-D49). Pieces from these puzzles are shown on the front cover of this hook. The base is a metallic looking plastic; the rest of the tray and the pieces are made of chunky plywood; the pieces are coloured according to shape and have rounded corners; and the puzzles are large enough to be a joy to manoeuvre. The only possible (very small) criticism is that there is no lid or box to put the puzzle in. In terms of handling, this series is far superior to anything else ever produced. Mr Ahe lives in the northernmost part of the Japanese mainland where he runs a place called 'Coffee Shop Now'. If he gives his puzzles, which are extremely difficult, to his customers while they are drinking coffee, he must sell coffee by the litre! It conjures up the picture of people sitting there weeks later still pushing little blocks of wood around, and having their 1000th cup of coffee!
..,
2
HISTORY OF THE SLIDING BLOCK
PUZZLE
History of the '15' and '14-15' puzzles
(A fuIllist of referencesis givenin AppendixB)
Who invented the sliding block puzzle (in pârticular the '15' Puzzle) and when? Unfortunately, authorities cannot agree on either.
During the latter part of the last century and the first part of this century, two people dominated the world of puzzles: Sam Loyd (1841-1911) in America and Henry Dudeney (1847-1930) in England. They each invented thousands of puzzles (of aIl kinds) and they also 'borrowed' heavily from each other. The same puzzles, in different guises, keep cropping up in their books. There is no doubt that Sam Loyd invented many puzzles, some of which became 'mini-crazes' in their own righl. But none had the world attention that was given the '15' Puzzle.
Dudeney (1926) maintains: 'But the great crazes only became possible under quite modern conditions. The first notable case was that of Loyd's 'fifteen' puzzle, that in 1873 was sold by the million and for a short time almost monopolized the attention of Europe and America . . . and the world positively "went mad" over this little thing'. Loyd himself (1914) says: 'The older inhabitants of Puzzleland will remember how in the early seventies 1 drove the entire world crazy over a little box of movable blocks which became known as the 14-15 Puzzle'.
But there is a wealth of evidence to show that not only was that date wrong but that Sam LQyd never invented il. If the above wording is looked at more dosely, he makes no daim on its invention--only that 'he drove the world crazy'. Many chronlders, especiaIly the later ones, assumed that Loyd invented it, presumably because of his long association with puzzles and inventing them.
Plate I. Clockwise trom top left: Rate Your Mind, Pal (Al); Four Square (AlO); La Grande Question (AS); Superpuzzle lOO-an example of an extremely difficult puzzle because ail the pieces are so nearly the same; Ten Little Nigger Boys (A6); The Premier (A2).
Plate II. Left to right: Au revoir/Do svidania (AB); Archie's Puzzle of the Heads (A9); Craps! (A9); Le Moulin Rouge (A7).
Plate III. Top row (left to right): Time Puzzle (B50); Parka Car (B35).
Bottom row: Vanish Mystery Puzzle (B29); Peyo Switchit (B34).
PlateV. Top row (left to right): Relax (C27); Mickeyand Minnie Compact
Puzzle (C22); Lost Pygmies(C69). Bottom Row: (Broken Heart) Reverse Puzzle (C27); Mintman Puzzle Mints (C27); Qwik-Sane (C3).
Plate IV. Clockwise from top left: Boss, Game of the Fifteen (B1O); Black
and White (B44); The New '15' Puzzle (B60); The So-Easy (B41); The Teaser (B42).
PlateVI. Clockwisefrom top left: Get My Goat (C2); Dad's puzzler (C19);
The Infants Hospital Puzzle (C12); George Washington Puzzle (C67h Kapture The Kron Prinz (C2).
;;;a",. . U
~
. ..,..
LI
...
m;.
\ ...
~."'"".: ~
-+0
o
o
~-~- -- -~---"--'~:-:" o. ___" . ' ~__Plate VII. Clockwise from top left: The Traffic Jam Puzzle (D12); Comic
Scramble Game (E20); The Klondyke Miners' Puzzle (E14); Ma's Puzzle (Dl); Climb Game 15 (D45).
Plate IX.. Three-dimensional sliding block puzzles. Front row (left to bottom): Qrazy Qube (G2); Varikon Box 'L (G3); Mad Marbles (G3). Back row (top to right): Change the Seasons (Gl); Pionir Cube (G4); Inversion (G3).
Platevrn. Top row (left to right): Orbit (E6); Manoeuvre (E30); Inter-City
(E29). Bottom row: Chifu-Chemulpo(E?); Automobile (EU); Perplexity
(ElO); Puzzle-Get Protection into . . . (F5).
PlateX. Top row (left to right}-various 3-D slidingpuzzles: Puzzle Pen; 6
by 6 (numbered); 6 by 6; Missing Link; Popeye Can Puzzle; Babylon Tower; Entrapment; 6 by 6 (with beads). Bottom row-various sliding puzzles with levers: 8-Pole Wisdom Plate; Row by Row; Rack 'em up; Great Gears.
Plate XI. Various sliding puzzles with plungers or levers. Top row (left to right): Hiroko; Four by Four; 5 x 5; Ten billion. Bottom row: Uriblock; Trillion; Crossover.
Plate XII. Railwayshunting puzzles.Top row (left to right): Turntable Train (113); Good Luck Railroad (17); Whistle Stop Puzzle (J4). Bottom row: Aiguillages (111); Shunting! (113); Pacific231 (13).
..--.
~~.~~
"~..7-
... ~'-.-.
1
IlHistory of the sliding block puzzle 19
Loyd and Dudeney made their name in puzzles partly because they were supreme at inventing and solving them; but an essential factor in their success was also their ability to 'dress up' an otherwise uninteresting mathematical exercise by embeIlishing it with stories and anecdotes. Many of the se were dreamed up for the purpose-especiaIly where it came to the origin of certain puzzles. ln his Eighth
book of Tan (1903, republished 1968), Loyd perpetrated an enormous
hoax on his readers about the origin of the Tangram. Unfortunately, even the eminent philologist, Sir James Murray was taken in, and went to considerable lengths only to find there was no basis in Loyd's story. It is best described by Peter van Note in his introduction to the Dover edition (Loyd 1968) of the same book.
1'"
He chose the title, he explained, because some four thousand years ago a Chinaman (or Chinese God) named Tan had compiled seven books of Tangram patterns. . . 'The seven books of Tan', said Loyd, 'were supposed to illustrate the creation of the world and the origin of the species upon a plan which out-Darwins Darwin, the progress of the human race being traced through seven stages of development up to a mysterious state which is too lunatic for serious consideration'. With that statement scholars like British philologist Sir James Murray should have been warned. ln his own special way Loyd blended half-truths and popular suppositionsabout things Oriental with a fewfigmentsof his own imagination; to this he added a few well-dropped names, and with a twinkle in his eye served the old puzzle up in a delightful mock-serious stew. Had Sir James looked more closely, he might have realized that Loyd's whole essay was 'too lunatic for serious consideration'. Instead Sir James embarked on a search for the historical Tan. . . (Loyd 1968). While the thought of China being scoured by one of Sir James' relatives on a wild goose chase is very amusing, it is far from helpful to the historian, and casts doubt on other statements of fact made by Loyd.
It is important to realize that there were two puzzles-not just one: the '14-15' Puzzle was quite different from the '15' Puzzle. The '15' puzzle has a 50 per cent chance of success-the '14-15' puzzle is impossible. Loyd (1914) further says: 'The fifteen blocks were arranged in the square box in regular order, only with the 14 and 15 reversed. The puzzle consisted in moving the blocks about, one at a time, so as to bring them back to the present position in every respect except that the error in the 14 and 15 must be corrected.'
This is quite different from the random placing of the blocks required for the '15' Puzzle, as described by aIl the early chroniclers.
20 Sliding Piece Puzzles
Johnson (1879): 'A ruled square of 16 compartments is numbered.
. .
Fifteen counters, numbered in like manner are placed at random upon squares so that one square is vacant.. .
The puzzle is to bring all the counters into their proper squares by successive moves'. Sala (1880): 'You take out the number 16; you mix up the counters in the box so that they will run irregularly'. Snowdon (1880): 'Sixteen small wooden cubes, numbered from one to sixteen, are placed in random order in a shallow box'. The instruction booklet entitled 'Albert Durer's Game of the Thirty-Four and Boss--Game of the Fifteen' produced by Cremer of 210 Regent Street, London in the spring of 1880 gives a long description and proof (?) of how 'only half the possible combinations can succeed'. This is clearly a reference to the random placing of pieces-not Loyd's 14-15. Kirkman (1880), Tait (1880), Warren (1880), Lucas (1881, 1882-94), and Schubert (1909) all refer to the random placing of pieces (i.e. the '15' Puzzle, not the '14-15 Puzzle). None of the above'mention Sam Loyd.Lucas, Schubert, and Ahrens are more specific and give dates as well. Lucas (1881) says that Mr Sylvester, the English mathematician, correspondent of the 'Academie des Science de Paris' and professor at the John Hopkins University of Baltimore, USA, told him the '15' puzzle had been invented by a deaf mute 18 months previously. The same article, with subtle amendments, was printed in the second edition of his book (Lucas 1882-1894), and one of the changes was the date. This now became 'towards the end of 1878'. Schubert (1909) gives the date as December 1878. Ahrens (1918) says that Sam Loyd is supposed to have invented the '15' puzzle hut also quotes the deaf mute story mentioned above and says that it is a legend. H e says the only certainty is that the puzzle was invented in America.
Loyd's son, Sam Loyd Jr., does not agree with his father about the date: 'It was in the early eighties when I had barely attained my 'teens that the world-disturbing "14-15 Puzzle" flashed across the horizon and Loyds were amongst its earliest victims. T o say that I was infatuated with the tantalizing box of blocks is a mild description of my enthralment' (Loyd 1928). Would he have written in this manner if his father had invented it?
The dates and the extent of the world's first big puzzle craze are fairly well documented. It started in America in 1879 and rapidly spread to Europe. The editors of the American Journal of Mathematics stated: 'The "Fifteen" puzzle for the last few weeks has been prominently before the American public, and may safely he said to have engaged the attention of nine out of ten persons of both sexes
History of the sliding blockpuzzle 21 and of all ages and conditions of the community' (Johnson 1879; Story 1879). Snowdon (1880) goes on to say: 'What then has caused this puzzle to become so immensely popular? How does it happen that one dealer in New York is said to have sold 230 gross (33 120) of a cheap variety of this puzzle in one day? Why is it that when once the puzzle is taken up no account is taken of the flight of time? That at the first attempt the figures may perhaps be soon shifted into the required consecutive order, while over the next random arrangement hours may he spent without the desired result being obtained?'
Proctor (1881b): 'It is singular to think that probably not fewer than twenty millions of persons tried The Boss Puzzle'.
Sala (1880):
A short Act of Parliament should be passed prohibiting, under penalty of heavy fine and long imprisonment, all and sundry of her Majesty's subjects from playing at a dreadful game called 'Fifteen' which is known in the United States as 'The Great Boss Puzzle'. . . But pshaw! what need have I to describe the fearsome game? Even as I write, thousands of my readers, old and young, may he playing it. If time he indeed money, that Great Boss Puzzle must have cost me at least a thousand dollars between January and June last. I played it at Omaha; I played it at Chicago; I played it at Great Salt Lake City; I played it on board the Hecla coming home; and upon my word, so soon as I have finished writing "Echoes", I shall be at the Great Bors Puzzle again. Why was it not stopped at the Custom-house? Why was it not brought under the provisions of the Dangerous Explosives or the Cattle Plague Laws? There would be no use in proceeding against the persons who have naturalised this appalling apparatus in England. Our old friend 'the merest schoolboy' can make a game of Fifteen for himself from so ma.ly buttons or draught-counters. It i s the players who, in the interests of Precious Time, should be punished. This passage in the Illustrated London News of 22 May 1880 gives us an accurate dating. Since Sala refers to 'between January and June last', it can only be interpreted as referring to 1879. This confirms other writers' dates of the craze starting in 1879 in America.
Lucas (1881) says that several months after its appearance in America the 'Jeu du Taquin' was imported into France and offered for sale in political and illustrated journals under the name of 'le double casse-t&te gaulois'. H e goes on to say that its success in Europe was perhaps even greater than in America.
Schubert (1909) says that the puzzle appeared in 1879 and 1880 in Germany under the title of 'Boss' puzzle, in England as the 'Fifteenth Puzzle' and in France as the 'Jeu du Taquin'. Ahrens (1918) says that
22 Sliding Piece Puzzles
it was introduced into Germany in 1878. H e also tells of Deputies in the Reichstag (Parliament) in 1880 not paying attention to the proceedings as they were too occupied with the Boss Puzzle.
Dudeney (1926): 'Certain London shops in Cheapside and elsewhere sold nothing else and were besieged from morning to night, while hawkers at every street found it impossible to supply the demand'.
They even wrote songs about it: 'THE FIFTEEN PUZZLE-sung with immense success by J. J . DALLAS in the Burlesque Drama "THE FORTY
THIEVES" at the Gaiety Theatre, (London) composed by GEORGE
M E E N , written by M R . REECE' (Punch, 8 January 1881; Meen and Reece 1881). Part
of
Mr Reece's song is reproduced on the dedication page at the front of this book.Loyd (1928): '. . . puzzle history's most notable event. .
.
The 14-15 puzzle craze did not come gradually..
. Instead, it burst upon our unsuspecting globe as might a meteor out of the sky. And the reverberations of its arrival spread with almost the speed of light to the uttermost corners of the world'.People became infatuated with the puzzle and ludicrous tales are told of shopkeepers who neglected to open their stores; of a distinguished clergyman who stood under a street lamp all through a wintry night trying to recall the way he had performed the feat. The mysterious feature of the puzzle is that no one seems to be able to recall the sequence of moves whereby they feel sure they succeeded in solving the puzzle. Pilots are said to have wrecked their ships; engineers rush their trains past stations and business generally became demoralised. A famous Baltimore editor tells how he went for his noon lunch and was discovered by his frantic staff long past midnight pushing little pieces of pie around on a plate. (Loyd 1914). If Loyd did not invent the '15' puzzle, then just what was his contribution? There seems little doubt that he added in no small way to its popularity. First, he decided on a particular arrangement of the pieces (instead of asking the solver to place them at random in the tray); and then he set the task of exchanging pieces 14 and 15, while, of course, ending up with all the other pieces in their original positions. H e might have chosen any two single pieces-the 1 and the 15, for example-because any such single exchange is impossible. Knowing that his puzzle was impossible he offered the enormous sum of $ 1000 to anyone who could provide a solution. 'A prize of $ 1000, which was offered for the first correct solution to the problem, has never been claimed, although therc are thousands of persons who say
History of the sliding blockpuzzle 23 they performed the required feat' (Loyd 1914). In an interview in 1907 Loyd again says:
'. .
.
in spite of that, however, there are thousands of persons in the United States who believe they solved that puzzle . . . but the thousand dollars reward I offered for anyone who could do it was never claimed. Not long ago the Sunday editor of a New York paper wanted to use it again as a supplement, and I suggested he should offer a thousand dollars reward for the solution. He refused. He said he remembered very well that he had done the puzzle once and he was not going to throw away a thousand dollars. Before I could persuade him to offer the reward, I had to bring the thousand dollars to his office and deposit it in the safe. It was never claimed.' (Bain 1907).Loyd must have achieved considerable publicity. Anyone who offered the equivalent today (it must be at least $ 25 000) would be assured of fairly extensive media coverage. A good recent example was that of Kit Williams and his Golden Hare, which achieved both national press and television news status.
What is not so clear is just when Loyd offered his reward. We can only conjecture that it was some time in 1880 or 1881. Proctor (1881~) wrote on 25 November 1881:
A prize is said to have been offered in America to anyone who should bring the blocks into this position-called the won position-starting from a position differing only from the 'won position' in having the three blocks in the fourth line arranged 13, 15, 14 instead of 13, 14, 15 (a position which has been called the 'lost.position'), and thousands wasted hours on hours of their time in the attempt to do this impossible thing. Some said they had done it, but were assuredly mistaken. Others thought they had satisfied the conditions of the problem by getting some such arrangements as. . . [Refer to puzzles B11, BIZ. (Chapter 4).]
Loyd claims that he tried to get a patent on the '14-15' Puzzle: MI Loyd has patented and copyrighted many of his inventions, but failed to get a patent on the 'Fourteen-Fifteen' puzzle. 'Of course, it couldn't be done', said Mr Loyd, 'and that's why I did not get my patent. It was necessary then to file with an application for a patent a "working model" of the device. When I applied for a patent, they asked me if it was possible to change the relations of the fourteen and fifteen. I said that it was mathematically impossible to do so'. 'Then', said the commissioner, 'you can't have a patent. For if the thing won't work, how can you file a working model of it?' 'His logic was alright, and the result was that I did not get my patent'. (Bain 1907).
24 Sliding Piece Puzzles
By 1881, it seems that the craze was almost over. On 11 November 1881, Proctor wrote: 'I thought the Fifteen puzzle was'dead, and hoped I had had some share in killing the time-absorbing monster. [It is an excellent puzzle by the way.
.
.]'
(Proctor 1881b).Much later Dudeney refers to the '14-15' Puzzle: 'And yet today it is dead as Queen Anne. Mathematicians set to work on the thing and discovered a rigid proof of its impossibility. When this became known it gave the thing a death blow, and the craze died as suddenly as it arose'. Dudeney (1926).
Whether this was true o r not for the '14-15' Puzzle, the public does not seem to have taken much notice of the mathematicians when it came to the earlier '15' Puzzle. The first proof-that half of all possible positions were solvable and half were not--came in 1879 in America from the American Journal of Mathematics (Johnson 1879; Story 1879). In 1880 the first proof appeared in the UK in the Royal Society of Edinburgh Proceedings (Tait 1880). Both these proofs also contained details of how to tell whether a random pattern was solvable or not. But the method proposed was rather laborious. It was simplified by Lucas, who in 1881 produced the first proof in France in the Revue Scientijique de la France et de r ~ t r a n g e r (Lucas 1881). The instruction leaflet to Albert Durer's Boss Puzzle by Cremer, already mentioned, has a proof of sorts. None of these proofs seems to have dented the popularity of the puzzle, which appears to have continued unabated until about the middle of 1881. The simplest way of determining whether a random pattern of the pieces is solvable or not is simply to count the number of exchanges necessary to get the pieces into their correct positions. If the count is even, it is solvable; if it is odd, it is impossible. With a little practice this can be done in one's head. It seems that this was readized (and even published) quite early on and a novel way of always solving the puzzle is suggested by Warren in The Nation:
1. Tip out all the pieces onto a table and make sure they are all face up. 2. Close your eyes and mix up all the pieces.
3. Put all the pieces hack into the box, still keeping your eyes closed. Naturally some pieces will face left, right or away from you.
4. Open your eyes and immediately count the exchanges necessary to solve the puzzle (the maximum is 14).
5. If the number is even rotate the individual pieces so that they all face you-and then solve the puzzle. However if the number is odd, first rotate the whole box 90" clockwise (or anticlockwise), then rotate the
individual pieces and solve the puzzle as before.
History of the sliding block puzzle 25
6 . As an added refinement the turning of the whole box can be avoided by turning instead piece 6 so that it becomes 9, and 9 so that it becomes 6. This will only he necessary when the wunt is odd and can he done in the course of righting all the other pieces. (Warren 1880).
Dozens of modern authors have all assumed that Loyd invented the '15' Puzzle. Few seem to have produced any evidence. Once two or three make the claim, it easily becomes self-perpetuating. It would be natural to think, from reading Loyd (1914) and Dudeney (1926), that Loyd had invented the puzzle in the early 1870s. Slocum (1986) is the most recent. H e aJds that he has a puzzle entitled: 'Embossed puzzle of Fifteen'-manufactured by the Embossing Co. which has printed on the lid, 'Patented Oct. 24, 1865'. The author has an identical puzzle, but Slocum confirms that a search at the U.S. Patent Office has revealed that no patent was issued to this company on that date.
Three things need to be remembered: (i) what was said by Loyd (1914) and Dudeney (1926) was published some thirty and forty years after the event; (ii) Sam Loyd's son does not agree with his father on the date (Loyd 1928) and writes in a way that makes it difficult to conclude that his father invented it; (iii) there is no contemporary evidence t o corroborate either a date of the early 1870s or that Loyd invented it.
On the contrary, there is ample evidence in the years 1879-81 to suggest that the '15' Puzzle (not the 14-1.5 puzzle) created a world- wide sensation during 1879 and 1880. Contemporary evidence also suggests that the 14-15 puzzle appearcd in 1881 (or possibly 1880). It is not certain whether Sam Loyd proposed the 14-15 version or someone else. But he did achieve notoriety by the offer of a large reward t o anyone who could solve it.
From all the information available, it would seem that the likely course of events was that the '15' Puzzle was invented in America towards the end of 1878 by an unknown person (not Sam Loyd). The resulting puzzle craze started in early 1879 and spread to Europe in the same year. Sometime later, in either 1880 or early 1881, Sam Loyd proposed his version, the '14-15' puzzle, which gained him immediate notoriety through the enormous reward offered for its solution. The craze then seems t o have died sometime in 1881.
The last word on the '15' Puzzle must go to the editors of the American Journal of Mathematics, who felt that they had to add some justification for the inclusion of the articles by Johnson and Story:
